"on" can mean to continue. Hence climb on & fly on can mean continue to climb & fly. "away" can be used similarly.

Yes. Climb on is used to invite someone to get on a bike or a horse or any object to be climbed onto. It is also a rock climbing sport expression to tell the climber they can proceed.

But "Can "Climb on" be an order to someone to get on -- a bicycle, for example, or a bus, or a boat.?

Thank you for your help!

#1 #2 and #4 work. I've never heard #3. Climb over implies getting over an obstacle such as 'Climb over the fence!" #5 - 7 work in that it is telling in what direction the bird should fly. But, fly as an imperative is used more as command to transition from not flying to flying. So, if I had a bird and it was sitting - I would say: 'Fly!' and while it is flying, I'd just say 'Up', 'down'. 'Fly over' is weird because it implies going over something. Like, fly over the lake. I'd probably just say: 'Fly over here'.